Sarnak has recently initiated the study of the Möbius function and its square, the characteristic function of square-free integers, from a dynamical point of view, introducing the Möbius flow and the square-free flow as the action of the shift map on the respective subshfits generated by these functions. In this paper, we extend the study of the square-free flow to the more general context of B-free integers, that is to say integers with no factor in a given family B of pairwise relatively prime integers, the sum of whose reciprocals is finite. Relying on dynamical arguments, we prove in particular that the distribution of patterns in the characteristic function of the B-free integers follows a shift-invariant probability measure, and gives rise to a measurable dynamical system isomorphic to a specific minimal rotation on a compact group. As a by-product, we get the abundance of twin B-free numbers. Moreover, we show that the distribution of patterns in small intervals of the form [N, N + √ N) also conforms to the same measure. When elements of B are squares, we introduce a generalization of the Möbius function, and discuss a conjecture of Chowla in this broader context.
We rephrase the conditions from the Chowla and the Sarnak conjectures in abstract setting, that is, for sequences in {−1, 0, 1} N * , and introduce several natural generalizations. We study the relationships between these properties and other notions from topological dynamics and ergodic theory.
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